Mathieu and Related Functions

Mathieu and related functions.

The Mathieu functions (RefLink[MathieuC, paclet:ref/MathieuC])[a,q,z] and (RefLink[MathieuS, paclet:ref/MathieuS])[a,q,z] are solutions to the equation y''+[a-2qcos(2z)]y==0. This equation appears in many physical situations that involve elliptical shapes or periodic potentials. The function (RefLink[MathieuC, paclet:ref/MathieuC]) is defined to be even in z, while (RefLink[MathieuS, paclet:ref/MathieuS]) is odd.

When q==0 the Mathieu functions are simply cos((Sqrt[a])z) and sin((Sqrt[a])z). For nonzero q, the Mathieu functions are only periodic in z for certain values of a. Such Mathieu characteristic values are given by (RefLink[MathieuCharacteristicA, paclet:ref/MathieuCharacteristicA])[r,q] and (RefLink[MathieuCharacteristicB, paclet:ref/MathieuCharacteristicB])[r,q] with r an integer or rational number. These values are often denoted by (Subscript[a, r]) and (Subscript[b, r]).

For integer r, the even and odd Mathieu functions with characteristic values (Subscript[a, r]) and (Subscript[b, r]) are often denoted c(Subscript[e, r])(z,q) and s(Subscript[e, r])(z,q), respectively. Note the reversed order of the arguments z and q.

According to Floquet's theorem, any Mathieu function can be written in the form (e^(irz))f(z), where f(z) has period 2π and r is the Mathieu characteristic exponent (RefLink[MathieuCharacteristicExponent, paclet:ref/MathieuCharacteristicExponent])[a,q]. When the characteristic exponent r is an integer or rational number, the Mathieu function is therefore periodic. In general, however, when r is not a real integer, (Subscript[a, r]) and (Subscript[b, r]) turn out to be equal.

This shows the first five characteristic values (Subscript[a, r]) as functions of q:

Plot[Evaluate[Table[MathieuCharacteristicA[r,q],{r,0,4}]],{q,0,15}]